An. Şt. Univ. Ovidius Constanţa
Vol. 17(2), 2009, 61–70
ON THE DETERMINANT OF AN
HODGE-DE RHAM LIKE OPERATOR
Mircea Craioveanu and Camelia-Ionela Petrişor
Abstract
Let (M, g) be a closed (i.e. compact without boundary), orientable,
n-dimensional smooth Riemannian manifold, C ∞ (M ) the real algebra of
smooth real functions on M , Ak (M ) the C ∞ (M )-module of smooth differential k-forms, 0 ≤ k ≤ n, and h a pointwise nonsingular tensor field
of type (1,1) on M with vanishing Nijenhuis tensor. For such h there
(k)
is an associated exterior derivative dh : Ak (M ) → Ak+1 (M ) having
(k+1)
an adjoint δh
: Ak+1 (M ) → Ak (M ) with respect to the usual global
inner product, so that one can define a (strongly) elliptic self-adjoint
(k)
second order differential operator ∆h : Ak (M ) → Ak (M ), which is a
generalization of the Hodge-de Rham operator (see [4]). In this paper
we shall discuss the smooth dependence on the Riemannian metric of
the determinant of this h-dependent Hodge-de Rham operator.
1
Introduction
Let (M, g) be a closed (i.e. connected, compact, and without boundary), oriented, n-dimensional smooth Riemannian manifold, C ∞ (M ) the real algebra
of smooth real functions on M and Ak (M ) the C ∞ (M )-module of smooth
differential k-forms, 0 ≤ k ≤ n. A tensor field h of type (1, 1) on M , which
can be conceived as a C ∞ (M )-linear mapping h : A1 (M ) → A1 (M ), induces
C ∞ (M )-linear mappings h(q) : Ak (M ) → Ak (M ) for any nonnegative integer
q, and the h(q) are defined by setting h(q) := 0 if q > k, and
h(q) (ω 1 ∧ . . . ∧ ω k ) :=
Key Words: h-dependent Hodge-de Rham operator, determinant of an h-dependent
Hodge-de Rham operator, smooth dependence
Mathematics Subject Classification: 58J50
Received: March, 2009
Accepted: september, 2009
61
62
=
Mircea Craioveanu and Camelia-Ionela Petrişor
X
1
sgn(σ)[h(ω σ(1) ) ∧ . . . ∧ h(ω σ(q) )] ∧ ω σ(q+1) ∧ . . . ∧ ω σ(k)
(k − q)!q!
σ∈Sk
if 0 ≤ q ≤ k, where ω i ∈ A1 (M ), 1 ≤ i ≤ k, σ runs through all permutations of {1, . . . , k}, and sgn(σ) denotes the sign of the permutation σ. The
transformation h(0) is taken to be the identity mapping on Ak (M ).
If h is a pointwise nonsingular tensor field of type (1, 1) on M with vanish(k)
ing Nijenhuis derivation, an h-dependent exterior derivation dh : Ak (M ) →
(k)
Ak+1 (M ) can be defined by setting dh := h(1) ◦ d(k) − d(k) ◦ h(1) , where
(k)
k
k+1
d
: A (M ) → A
(M ) is the usual exterior differential. When h is the
(k)
identity, dh coincides with d(k) . Let δ (k+1) : Ak+1 (M ) → Ak (M ) be the ad(1)
joint of d(k) relative to the usual inner product induced by g. If ht denotes
(k+1)
the transpose of h(1) , the mapping δh
: Ak+1 (M ) → Ak (M ), defined by
(k+1)
(1)
(1)
(k)
(k+1)
(k+1)
setting δh
:= δ
◦ ht − ht ◦ δ
, is the adjoint of dh with respect to the usual inner product. The h-dependent Hodge-de Rham operator
(k+1)
(k)
(k)
(k)
(k−1)
(k)
◦ δh + δh
◦ dh ,
∆h : Ak (M ) → Ak (M ), defined by setting ∆h := dh
is elliptic and self-adjoint, and it reduces to the usual Hodge-de Rham operator
∆(k) in the case when h is the identity (see [4]).
2
Spectral properties of the h-dependent
Hodge-de Rham operators
In what follows we assume that h is a nonsingular tensor field of type (1, 1) on
(k)
M with vanishing Nijenhuis derivation. Since ∆h : Ak (M ) → Ak (M ) is an
elliptic second order differential operator, formally self-adjoint and formally
positive, and (M, g) is a closed, smooth Riemannian manifold, the following
statements are valid (see also [2]).
Theorem 2.1 (i) For each k ∈ {0, 1, . . . , n}, there exists a discrete spectral
(k)
(k)
(k)
resolution {ωh;j , λh;j }j∈N of the operator ∆h : Ak (M ) → Ak (M ), where
(k)
(k)
ωh;j ∈ Ak (M ) for any j ∈ N, that is {ωh;j }j∈N is a complete orthonormal system in the real Hilbert space L2 (Ak (M )) = W 0,2 (Ak (M )) such that
(k) (k)
(k) (k)
(k)
∆h ωh;j = λh;j ωh;j for any j ∈ N. Moreover, λh;j ∈ [0, +∞) for any j ∈ N,
(k)
each eigenspace of ∆h is finite dimensional, and 0 ∈ R is an eigenvalue of
(k)
∆h : Ak (M ) → Ak (M ) if and only if the k-th Betti number βk (M ) 6= 0;
(ii) For any m ∈ N , there exists ℓ(m) ∈ N∗ and a constant Cm > 0, such
that
(k)
(k)
||ωh;j ||∞,m ≤ Cm (1 + (λh;j )ℓ(m) ),
63
ON THE DETERMINANT OF AN HODGE-DE RHAM LIKE OPERATOR
where
(k)
(k)
||ωh;j ||∞,m := sup |jm (ωh;j )x |J m (Λk T ∗ M )x ,
x∈M
(k)
jm (ωh;j )x denoting the m-jet of the smooth section ωh;j ∈ C ∞ (Λk T ∗ M ) at the
point x ∈ M ;
(k)
(iii) If one arrange the eigenvalues of ∆h such that
(k)
(k)
0 ≤ λh;0 ≤ λh;1 ≤ . . . ,
(k)
then there exist real constants C(k) > 0 and ε(k) > 0 such that λh;j ≥
C(k)j ε(k) if j ≥ j0 is sufficiently large;
(k)
(k)
(iv) Let aj := hθ, ωh;j i ∈ R, j ∈ N, be the Fourier coefficients associated
to θ ∈ L2 (Ak (M )). If θ ∈ Ak (M ), then
X (k) (k)
|aj |λh;j < +∞
j∈N
P
(k) (k)
and the series j∈N |aj |λh;j tends to θ uniformly with respect to the norm
k · k∞,i for any k ∈ {0, 1, . . . , n}.
For an initial smooth differential k-form θ ∈ Ak (M ), let us consider the
(k)
heat equation associated to ∆h : Ak (M ) → Ak (M ) with the initial condition
θ:
∂
(k)
(1)
( + ∆h )ω(x, t) = 0,
∂t
lim ω(x, t) = θ(x),
(2)
tց0
(k)
where x ∈ M , t ∈ (0, +∞). Let (e−t∆h )(θ) be the unique solution of the evolution equation (1) that satisfies the initial condition (2). The linear operator
(k)
e−t∆h : Ak (M ) → Ak (M ) can be extended to a compact, self-adjoint opera(k)
(k)
(k)
tor from L2 (Ak (M )) into L2 (Ak (M )), denoted also by e−t∆h . Let {ωh;j , λh;j }
(k)
be a discrete spectral resolution of ∆h : Ak (M ) → Ak (M ) and let us consider
the Fourier expansion of θ ∈ Ak (M ):
X (k) (k)
(k)
(k)
aj ωh;j , where aj := hθ, ωh;j i, j ∈ N.
θ=
j∈N
∗(k)
∗(k)
(k)
Let us define ωh;j ∈ (Ak (M ))∗ by ωh;j (̟) := h̟, ωh;j i, ̟ ∈ Ak (M ),
∗(k)
(k)
such that kωh;j k = kωh;j k. With these notations, the following statement is
valid.
64
Mircea Craioveanu and Camelia-Ionela Petrişor
Theorem 2.2 The following two series converge uniformly in the C ℓ -topology
for any ℓ ∈ N if t ≥ δ > 0:
(k)
Eh (x, y, t) :=
X
(k)
(k)
∗(k)
e−tλh;j ωh;j (x) ⊗ ωh;j (y) ∈ Hom(Λk (Ty∗ M ), Λk (Tx∗ M ))
j∈N
and
(k)
−t∆h
(e
(θ))(x, t) :=
X
(k)
(k) (k)
e−tλh;j aj ωh;j (x)
:=
Z
M
j∈N
(k)
(Eh (x, y, t))θ(y)dµg (y),
where x, y ∈ M , and µg denotes the canonical measure on M associated to g.
Let us define the matrix of Fourier coefficients of the bounded linear operator
(k)
e−t∆h : L2 (Ak (M )) → L2 (Ak (M ))
by
(k)
(k)
(k)
(k)
aij (Eh ) : = he−t∆h (ωh;i ), ωh;j i =
Z
Z
(k)
(k)
(k)
=
(Eh (x, y, t))(ωh;i (y)|ωh;j (x))dµg (x)dµg (y),
x∈M
y∈M
i, j ∈ N, k ∈ {0, 1, . . . , n}, where ( | ) denotes the pointwise inner product on
Ak (M ) defined by using the fiber metric on Λk T ∗ M induced by g.
Theorem 2.3 (i) With the previous notations the following equalities
(k)
T raceL2 (e−t∆h ) : =
Z
(k)
x∈M
=
X
T raceΛk (Tx∗ M ) (Eh (x, x, t))dµg (x) =
(k)
ajj (Eh;j ) =
X
(k)
e−tλh;j
j∈N
j∈N
are valid for each k ∈ {0, 1, . . . , n}, that is the continuous linear operator
(k)
e−t∆h : L2 (Ak (M )) → L2 (Ak (M )) is a trace class operator;
(ii) If (A(M ), dh ) is the elliptic cochain complex previously defined, then
n
X
(k)
(−1)k T raceL2 (e−t∆h ) = Index(A(M ), dh ) = χ(M ),
k=0
where χ(M ) denotes the Euler-Poincaré characteristic of M .
ON THE DETERMINANT OF AN HODGE-DE RHAM LIKE OPERATOR
65
(k)
Finally, let us discuss the asymptotics of T raceL2 (e−t∆h ) as t ց 0 and
relate these asymptotics to Index(A(M ), dh ).
(k)
For each t ∈ (0, 1) and k ∈ {0, 1, . . . , n}, the restriction of Eh (·, ·, t) on
the diagonal of M × M admits an asymptotic expansion. More precisely, the
following statement is valid.
Lemma 2.4 For each k ∈ {0, 1, . . . , n} and each m ∈ N, there exists a smooth
(k)
mapping em (·, ∆h ) : M → End(Λk (T ∗ M )) such that
(k)
(i) em (x, ∆h ) ∈ End(Λk (Tx∗ M )) depends functorially on a finite number
(k)
of jets of the symbol of the second order differential operator ∆h ;
P
(k)
(k) m−n
(ii) Eh (x, x, t) ∼ m em (x, ∆h )t 2 as t ց 0, for any x ∈ M ;
(k)
(iii) em (x, ∆h ) = 0 for any x ∈ M and any m odd.
For the proof, we refer to [5], Lemma 1.8.2.
Let
(k)
(k)
M ∋ x 7→ am (x, ∆h ) := T race(en (x, ∆h )) ∈ R
and
(k)
am (∆h )
:=
Z
x∈M
(k)
am (x, ∆h )dµg (x) ∈ R,
(3.3)
the invariant scalar functions and the numerical invariants respectively asso(k)
ciated to the differential operator ∆h , k ∈ {0, 1, . . . , n}.
(k)
(k)
Lemma 2.5 (i) The invariants em (x, ∆h ) and am (x, ∆h ), m even, can be
expressed by local formulas that are homogeneous of order m in the jets of the
(k)
total symbol of ∆h for each k ∈ {0, 1, . . . , n};
(k)
(ii) am (x, ∆h ) = 0 for each m odd and any k ∈ {0, 1, . . . , n};
P
(k)
(k) m−n
(iii) T raceL2 (e−t∆h ) ∼ m am (∆h )t 2 as t ց 0, for each k ∈ {0, 1, . . . , n}.
For the proof of statement (i) we refer to [5], Lemma 1.8.3(c), while the
statement (ii) is an immediate consequence of Lemma 2.4(iii). Statement
(k)
(iii) is an immediate consequence of the definition of T raceL2 (e−t∆h ) [see
Theorem 2.3(i)] and of Lemma 2.4(ii).
3
The determinant of an h-dependent
Hodge-de Rham operator
In this Section we assume that the k-th Betti number βk (M ) of M is equal
to zero, so we do not must to deal with the 0-spectrum. Let R be a complementary region to a cone about positive real axis lying in the right half plane
66
Mircea Craioveanu and Camelia-Ionela Petrişor
(k)
C, cone that includes the symbolic spectrum of ∆h , so that the intersection
of this symbolic spectrum with R is empty. We normalize the choice of R so
that the boundary γ is a curve about the positive real axis which consists of
a portion of a large circle around the origin and of two rays lying in the right
half plane. We orient γ in a clockwise fashion. If λ ∈ R and |λ| is large,
(k)
(k)
then (∆h − λ) is invertible. Since ∆h is positive definite, we may assume
Re(λ) > 0 for λ ∈ γ. Choose the branch of λs , s ∈ C, defined on the right
(k)
half plane so 1s = 1. The L2 -norm of the operator (∆h − λ)−1 is uniformly
2
k
bounded in L (A (M )). For Re(s) > 1, the integral
Z
(k)
(k)
(∆h )−s := (2πi)−1 λ−s (∆h − λ)−1 dλ
γ
converges to a bounded operator on L2 (Ak (M )).
(k)
(k)
(k)
Let {ωh;j , λh;j }j∈N be a discrete spectral resolution of ∆h [see Theorem
2.1(i)], and
X (k)
(k)
(k)
(k)
∗(k)
Eh (x, y, s, ∆h ) :=
(λh;j )−s ωh;j (x) ⊗ ωh;j (y)
(3)
j
(k)
the kernel of (∆h )−s . Theorem 2.1 implies the existence of real constants
δ > 0 and ℓ(m) > 0 such that
(k)
(k)
(k)
λh;j ≥ j δ and ||ωh;j ||∞,m ≤ (λh;j )ℓ(m) ,
(4)
whence it follows that (3) converges absolutely and uniformly in the || · ||∞,m norm, for Re(s) large.
Definition 3.1 We define the generalized zeta function
X (k)
(k)
(k)
ζ(s, ∆h ) := traceL2 ((∆h )−s ) =
(λh;j )−s =
j
=
Z
(k)
M
(k)
T race(Eh (x, x, s, ∆h ))dµg (x).
Using relations (4), one can show that it converges uniformly and absolutely
for Re(s) large.
We use the Mellin transform to relate the zeta function to the heat kernel.
From Lemma 2.5(iii), Theorem 2.3(i) and the identity
Z ∞
(k)
(k)
ts−1 e−λh t dt = (λh )−s Γ(s),
0
where Γ : {s ∈ C| Re(s) > 0} → C, Γ(s) :=
following theorem.
R∞
0
ts−1 e−t dt, we obtain the
ON THE DETERMINANT OF AN HODGE-DE RHAM LIKE OPERATOR
67
Theorem 3.2 (i) If Re(s) is large, then
Z ∞
(k)
(k)
−1
ts−1 T raceL2 (e−t∆h )dt;
ζ(s, ∆h ) = Γ(s)
0
(k)
(ii) Γ(s)ζ(s, ∆h ) has
poles at s = n−m
for m ∈
2
a meromorphic extension to C with isolated simple
N and
(k)
(k)
Ress= n−m Γ(s)ζ(s, ∆h ) = am (∆h );
2
(k)
(iii) Let ∆h (ǫ) be a smooth one parameter family of such operators. Then
(k)
ζ(s, ∆h (ǫ)) is smooth with respect to (s, ǫ) away from the exceptional values
at s = n−m
2 .
Remark 3.3 Because by the determinant of the h-dependent Hodge-de Rham
(k)
operator ∆h we understand the product of all eigenvalues, considered with
their multiplicity, that is
(k)
det(∆h ) =
∞
Y
(k)
λh,j ,
j=1
by Theorem 3.2 one can define a regularization of it, using the generalized zeta
function, namely
(k)
− d ζ(s,∆h )
(k)
detζ (∆h ) = e ds s=0
.
(5)
Note that this is formally correct since if there were only a finite number
of eigenvalues we would have


X (k)
d X (k) −s 
(k)
(λh,j )
=−
(λh,j )−s log(λh,j )
ds
j
j
Q (k)
which takes the value − log( j λh,j ) at s = 0.
In what follows, we endow the set M(M ) of all smooth Riemannian metrics
on M with the structure of a Fréchet manifold (see [1] and [7]). In the paper
[6] it is proven the following result.
Corollary 3.4 If M is a closed, n-dimensional smooth manifold, and h a
nonsingular tensor field of type (1, 1) on M with vanishing Nijenhuis derivation, then for each fixed t ∈ (0, ∞), the real functions
(k)
T raceL2 (e−t∆h
(·)
)=
∞
X
j=1
(k)
eλh,j (·)t : M(M ) → R
68
Mircea Craioveanu and Camelia-Ionela Petrişor
are smooth with respect to the canonical Fréchet manifold structure considered
on M(M ).
Therefore, by relation (5), by Theorem 3.2 and by Corollary 3.4 one obtains
Corollary 3.5 If M is a closed, n-dimensional smooth manifold, and h a
nonsingular tensor field of type (1, 1) on M with vanishing Nijenhuis derivation, then for each fixed t ∈ (0, ∞), the real functions
(k)
ζ(s,∆h (·))
− d (k)
detζ (∆h (·)) = e ds s=0
: M(M ) → R
are smooth with respect to the canonical Fréchet manifold structure considered
on M(M ).
References
[1] E. Binz, J. Śniatycki and H. Fischer, Geometry of Classical Fields, Mathematical Studies 154, North Holland Publishing, Amsterdam, 1989.
[2] M. Craioveanu, V.-D. Lalescu and C.-I. Petrişor, Spectral properties of a
class of Hodge-de Rham like operators, Analele Ştiinţifice ale Universităţii
”Al.I.Cuza” din Iaşi (S.N.) Matematică, Tomul LIII, 2007, Supliment,
107-122.
[3] M. Craioveanu, M. Puta and Th.M. Rassias, Old and New Aspects in
Spectral Geometry, Springer-Verlag, Mathematics and Its Applications,
Vol.534, Berlin, Heidelberg, New York, 2001.
[4] P.R. Eiseman and A.P. Stone, A generalized Hodge theory, J. Differential
Geometry, Vol.9 (1974), 169-176.
[5] P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer
Index Theorem, Second Edition, CRC Press, Boca Raton, Ann Arbor,
London, Tokyo, 1995.
[6] C.-I. Petrişor, Properties of the eigenvalues of certain elliptic differential
operators, Contemporary Geometry and Topology and Related Topics,
Dorin Andrica, Sergiu Moroianu (Editors), Cluj University Press, 2008,
237-242.
[7] J. Wenzelburger, On the smooth deformation of Hilbert space decompositions, Appendix in G. Schwarz, Hodge Decomposition-A Method for
Solving Boundary Value Problems, Springer-Verlag, Berlin, Heidelberg,
New-York, 1995.
ON THE DETERMINANT OF AN HODGE-DE RHAM LIKE OPERATOR
West University of Timişoara
Department of Mathematics
B-dul V. Pârvan, Nr. 4, 300223-Timişoara, România
Email: craioveanumircea@yahoo.com
”Politehnica” University of Timişoara,
Piaţa Victoriei, Nr. 2, 300006-Timişoara, România
Email: camelia.petrisor@mat.upt.ro
69
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Mircea Craioveanu and Camelia-Ionela Petrişor